Keith Davidson, the ex-lawyer for former adult-film actress Stephanie Clifford, denied Thursday that he had colluded with a lawyer for President Donald Trump over a hush-money deal, arguing that he tried to fulfill her goal of making money off an alleged sexual encounter with Mr. Trump.

Ms. Clifford sued Mr. Davidson in Los Angeles County court on Wednesday, accusing him of being a “puppet” for Mr. Trump and the president’s personal lawyer, Michael Cohen.